Problem: Simplify and expand the following expression: $ \dfrac{2}{2t - 10}+ \dfrac{5}{t - 5}- \dfrac{4t}{t^2 - 10t + 25} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{2}{2t - 10} = \dfrac{2}{2(t - 5)}$ We can factor the quadratic in the third term: $ \dfrac{4t}{t^2 - 10t + 25} = \dfrac{4t}{(t - 5)(t - 5)}$ Now we have: $ \dfrac{2}{2(t - 5)}+ \dfrac{5}{t - 5}- \dfrac{4t}{(t - 5)(t - 5)} $ The least common multiple of the denominators is: $ 2(t - 5)(t - 5)$ In order to get the first term over $2(t - 5)(t - 5)$ , multiply by $\dfrac{t - 5}{t - 5}$ $ \dfrac{2}{2(t - 5)} \times \dfrac{t - 5}{t - 5} = \dfrac{2(t - 5)}{2(t - 5)(t - 5)} $ In order to get the second term over $2(t - 5)(t - 5)$ , multiply by $\dfrac{2(t - 5)}{2(t - 5)}$ $ \dfrac{5}{t - 5} \times \dfrac{2(t - 5)}{2(t - 5)} = \dfrac{10(t - 5)}{2(t - 5)(t - 5)} $ In order to get the third term over $2(t - 5)(t - 5)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4t}{(t - 5)(t - 5)} \times \dfrac{2}{2} = \dfrac{8t}{2(t - 5)(t - 5)} $ Now we have: $ \dfrac{2(t - 5)}{2(t - 5)(t - 5)} + \dfrac{10(t - 5)}{2(t - 5)(t - 5)} - \dfrac{8t}{2(t - 5)(t - 5)} $ $ = \dfrac{ 2(t - 5) + 10(t - 5) - 8t} {2(t - 5)(t - 5)} $ Expand: $ = \dfrac{2t - 10 + 10t - 50 - 8t}{2t^2 - 20t + 50} $ $ = \dfrac{4t - 60}{2t^2 - 20t + 50}$ Simplify: $ = \dfrac{2t - 30}{t^2 - 10t + 25}$